Thermal management devices such as heat exchangers and heat pipes are integral to safe and efficient performance in multiple engineering applications, including lithium-ion batteries, electric vehicles, electronics, and renewable energy. However, the functional designs of these devices have until now been created around conventional manufacturing constraints, and thermal performance has plateaued as a result. While 3D printing offers the design freedom to address these limitations, there has been a notable lack in high thermal conductivity materials beyond aluminium alloys. Recently, the 3D printing of pure copper to sufficiently high densities has finally taken off, due to the emergence of commercial-grade printers which are now equipped with 1 kW high-power lasers or short-wavelength lasers. Although the capabilities of these new systems appear ideal for processing pure copper as a bulk material, the performance of advanced thermal management devices are strongly dependent on topology-optimised filigree structures, which can require a very different processing window. Hence, this article presents a broad overview of the state-of-the-art in various additive manufacturing technologies used to fabricate pure copper functional filigree geometries comprising thin walls, lattice structures, and porous foams, and identifies opportunities for future developments in the 3D printing of pure copper for advanced thermal management devices.